Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory

TL;DR

This paper provides an exact kinetic theory-based solution to the two-body stochastic dealer model, clarifying its dynamics and order-book profile, and extends it to include trader interactions, supported by numerical validation.

Contribution

It offers the first exact solution for the two-body dealer model using kinetic theory, enhancing understanding of microscopic financial trader interactions.

Findings

Exact master-Liouville equations derived for N=2

Order-book profile and transaction intervals obtained analytically

Extended model with trader interactions solved exactly

Abstract

The two-body stochastic dealer model is revisited to provide an exact solution to the average order-book profile using the kinetic approach. The dealer model is a microscopic financial model where individual traders make decisions on limit-order prices stochastically and then reach agreements on transactions. In the literature, this model was solved for several cases: an exact solution for two-body traders $N=2$ and a mean-field solution for many traders $N\gg 1$. Remarkably, while kinetic theory plays a significant role in the mean-field analysis for $N\gg 1$, its role is still elusive for the case of $N=2$. In this paper, we revisit the two-body dealer model $N=2$ to clarify the utility of the kinetic theory. We first derive the exact master-Liouville equations for the two-body dealer model by several methods. We next illustrate the physical picture of the master-Liouville equation from the viewpoint of the probability currents. The master-Liouville equations are then solved exactly to derive the order-book profile and the average transaction interval. Furthermore, we introduce a generalised two-body dealer model by incorporating interaction between traders via the market midprice and exactly solve the model within the kinetic framework. We finally confirm our exact solution by numerical simulations. This work provides a systematic mathematical basis for the econophysics model by developing better mathematical intuition.